A lifting theorem for operators between Lipschitz spaces

Abstract

We prove that every bounded linear operator between Lipschitz spaces admits a lifting along the de Leeuw embedding. More precisely, given pointed metric spaces $M$ and $N$ and $\hat{I}\mu >0$, every bounded linear operator $S:{\mathrm{Lip}}_0(M)\to {\mathrm{Lip}}_0(N)$ admits a lifting $\mathfrak{S}:C({\hat{I}}^2\tilde{M})\to C({\hat{I}}^2\tilde{N})$ such that $\Vert \mathfrak{S}\Vert \le \Vert S\Vert +\hat{I}\mu$ and $\mathfrak{S}({\Phi }_M(f))={\Phi }_N(S(f))$ for every $f\in {\mathrm{Lip}}_0(M)$. Moreover, compact operators admit compact liftings.